#19089

order	256
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	Aut(G) = C₂×#72286730 (order 4096) · Inn(G) = C₂×#120 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

425 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,23,28)(19,29,31,41)(20,33,35,42)(21,27,22,26)(24,36,39,32)(25,37,40,34)(30,38,44,48)(43,45,47,46);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,41)(27,29)(28,39)(33,47)(34,46)(35,44)(37,45)(40,48)(42,43);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20,23,35)(18,25,28,40)(19,30,31,44)(21,34,22,37)(24,38,39,48)(26,33,27,42)(29,43,41,47)(32,45,36,46);(2,7)(6,15)(8,12)(9,16)(19,31)(24,39)(29,41)(30,44)(32,36)(38,48)(43,47)(45,46)

Polycyclic presentation:

2^8|a^2=g,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#250488, #250493, #250496, …
maximal normal subgroup of	#250488, #250493, #250496, …
maximal quotient of	#168912, #168926, #168932, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12077);